Dealing with Uncertainty in Measurement
How do we deal with this uncertainty, besides pretending it does not exist? In the most ordinary measuring of things, one reasonably careful measurement is all that is needed. Most people will settle for that, believing that they know the length of a thing, or its time, or its weight, and so forth. However, many scientific and technical workers must work as nearly as they can to the limits of the accuracy of their instruments and methods. They must also report to others what those limits of accuracy are. That is why there is need for ways to express these limits.
Here is one method: Let's say that you are measuring across a piece of paper with a scale such as a ruler divided into centimeters (cm) and tenths of a centimeter (or millimeter). You want to express your measurement as centimeters to the nearest tenth, such as 23.7 cm. (See figure 10.2.)
As you measure, you will see that the edge of the paper is not quite on one of the marks showing tenths, so pick the tenth mark that the edge is nearest to. That is, pick 23.7, not 23.6. (See figure 10.3.)
However, you do not want anyone to think that you are claiming that the paper measured exactly 23.7 cm. Instead, you would like to tell others that you are reading the scale to within onehalf of a division either way from a scale mark. (See figure 10.4.)
Onehalf of onetenth is 0.05 in decimal notation. This is why you put "± 0.05 cm" after your measurement figure. For example: 23.7 ± 0.05 cm (which reads, "23.7 plus or minus fivehundredths of a centimeter").
For another method of expressing uncertainty, let's go back to the measurement of the width of a room. There we made five measurements and reported the mean as 5.828 meters. If we had wanted to express the same measurement in millimeters (mm) instead of in meters, we could have reported 5,828 mm. Remember, 5,828 mm is the average of five measurements of which the shortest was 5,823 mm and the longest was 5,834 mm. This spread from lowest to highest we call the "range" of the 5 measures. We express the range as 11 mm by subtracting the shortest measurement from the longest.
Our mean of 5,828 mm is 6 mm below the top of the range and 5 mm above the bottom. We may also want people to know the range when we report the mean of several measurements. This is recorded as: 5,828 ± 6 mm. Or, we could report it in its meter unit: 5.828 ± 0.006 m.
This system, using the mean to express the range, is an overly simplified explanation of the method, but it does show how the system works.
There is yet another way to express the uncertainty of measurements. As a simple example, let's say that you are measuring a rod with a meter stick and find that it is just under 1 meter. That is reported as 0.999 m (or 999 mm) because you wish to report that the measurement is nearer to 0.999 m than to 1.000 m (or nearer to 999 mm than to 1,000 mm). You are saying that the range is within 1 mm out of about 1,000 mm. Therefore, you report your measurement as 0.999 m with an "error" of 1 part per 1,000. The term "error" is not used in this method to mean that you made a mistake; it is another way of expressing the usual inaccuracy of any measurement.
Suppose, however, that you take the same meter stick and measure a room that is close to 10 m (or 10,000 mm) across. You find that several measurements range from 9,995 to 10,005, or a net range of 10 mm. This, too, shows an error of about 10 parts in 10,000 or 1 part in 1,000, as in the previous example.
To say that these measurements are accurate to 1 part in 1,000, we must assume that your meter stick is "exactly" 1 meter long. Of course, we cannot claim that because we are working on the assumption that no two things are exactly alike (including your meter stick and some master meter stick).
All measuring instruments, including wooden meter sticks, do change from their original dimensions. This unstable nature of things is part of our measurement problem. The changes that come with variations in temperature, humidity, and other factors must always be expected as part of our measuring and should be allowed for in our reporting if we are trying for high accuracy. Scientists working on one of the most important measuring problems, the measurement of the speed of light, using the best equipment and methods they can devise, must still allow for the uncertainty of their measurements. Thus, they report the velocity of light in a vacuum as 299,792,456 ± 1.1 meters per second.
We have discussed three different ways of reporting the uncertainty of measurements: (1) Reference to onehalf of the smallest scale division; (2) Reference to a range of several values above and below the mean of those values; (3) The error in parts per 1,000, parts per million, or the like. Now that you better understand how the "pros" work with uncertainty in measurement, you may be able to handle it fairly easily while you do your own science investigations.
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